Question
The geometric mean of $u$ and $v$ is $G=\sqrt{u v}$ and the arithmetic mean is $A=(u+v) / 2$ Show that if $u=x, v=x+c, c$ a real number, then$$\frac{d G}{d x}=\frac{A}{G}.$$
Step 1
We can find the geometric mean $G$ and the arithmetic mean $A$ using these expressions: $$G = \sqrt{uv} = \sqrt{x(x+c)} = \sqrt{x^2 + cx}$$ $$A = \frac{u+v}{2} = \frac{x + (x+c)}{2} = \frac{2x+c}{2}$$ Now, we want to find the derivative of $G$ with respect to Show more…
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